But setting up the integral is the only interesting part of this problem!

Since this involves an upside down cone, it's probably best to use cylindrical coordinates (if it had been a "regular" cone, spherical coordinates). In cylindrical coordinates the sphere is given by, of course, and the cone is given by z= 4- r. The differential of volume is . The cone and the sphere intersect when or . That is, they intersect at r= 0 (the top of the sphere) and at r= 4, the circumference.

To cover the entire sphere, r must go from 0 to 2 and must go from 0 to . For each r, z must go from the cone z= 4- r up to the upper half of the sphere, z= \sqrt{16- r^2}.